#197189

order	512
nilpotency class	3
isoclinism	isoclinic class of #197026
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60802	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1099 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,8,5,9,12,6)(3,7,11,15,10,14,16,13)(17,18)(19,28)(20,32,35,42)(21,36)(22,26)(23,27)(24,31)(25,34,41,33)(29,44,45,38)(30,40)(37,39)(43,46,48,47);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,21,31,23,30,22,37)(18,24,26,28,27,39,36,40)(20,29,34,47,35,45,33,46)(25,38,42,43,41,44,32,48);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,35)(24,38)(26,42)(27,41)(28,43)(30,45)(31,47)(32,36)(37,46)(39,44)(40,48);(3,10)(7,14)(11,16)(13,15)(19,30)(24,39)(28,40)(29,45)(31,37)(38,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 104385);